[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Effective simulation of state distribution in qubit chains

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

This work recollects a non-universal set of quantum gates described by higher-dimensional Spin groups. They are also directly related with matchgates in theory of quantum computations and complexity. Various processes of quantum state distribution along a chain such as perfect state transfer and different types of quantum walks can be effectively modeled on classical computer using such approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Valiant, L.G.: Quantum computers that can be simulated classically in polynomial time. In: Proceedings of 33rd Annual ACM STOC, pp. 114–123 (2001)

  2. Terhal, B.M., DiVincenzo, D.P.: Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A 65, 032325 (2002)

    Article  ADS  Google Scholar 

  3. Knill, E.: Fermionic linear optics and matchgates (2001). Preprint arXiv:quant-ph/0108033

  4. Kitaev, A.Y.: Unpaired Majorana fermions in quantum wires. Phys.-Usp. (Suppl.) 44, 131–136 (2001)

    Article  ADS  Google Scholar 

  5. Bravyi, S., Kitaev, A.: Fermionic quantum computation. Ann. Phys. 298, 210–226 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  6. Vlasov, A.Y.: Clifford algebras and universal sets of quantum gates. Phys. Rev. A 63, 054302 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  7. Jozsa, R., Miyake, A.: Matchgates and classical simulation of quantum circuits. Proc. R. Soc. Lond. Ser. A 464, 3089–3106 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  8. Jozsa, R., Kraus, B., Miyake, A., Watrous, J.: Matchgate and space-bounded quantum computations are equivalent. Proc. R. Soc. Lond. Ser. A 466, 809–830 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  9. Jozsa, R., Miyake, A., Strelchuk, S.: Jordan–Wigner formalism for arbitrary 2-input 2-output matchgates and their classical simulation. Quantum Inf. Comput. 15, 541–556 (2015)

    MathSciNet  Google Scholar 

  10. Brod, D.J.: Efficient classical simulation of matchgate circuits with generalized inputs and measurements. Phys. Rev. A 93, 062332 (2016)

    Article  ADS  Google Scholar 

  11. Vlasov, A.Y.: Quantum gates and Clifford algebras (1999). Preprint arXiv:quant-ph/9907079

  12. Wilczek, F.: Majorana returns. Nat. Phys. 5, 614–618 (2009)

    Article  Google Scholar 

  13. Nayak, C., Simon, S., Stern, A., Freedman, M., Das, Sarma S.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  14. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)

    Article  ADS  Google Scholar 

  15. Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33rd Annual ACM STOC, pp. 60–69 (2001)

  16. Kempe, J.: Quantum random walks—an introductory overview. Contemp. Phys. 44, 307–327 (2003)

    Article  ADS  Google Scholar 

  17. Kendon, V.: Decoherence in quantum walks—a review. Math. Struct. Comput. Sci. 17(6), 1169–1220 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11(5), 1015–1106 (2012)

    Article  MathSciNet  Google Scholar 

  19. Barenco, A., Deutsch, D., Ekert, A.K., Jozsa, R.: Conditional quantum dynamics and logic gates. Phys. Rev. Lett. 74, 4083–4086 (1995)

    Article  ADS  Google Scholar 

  20. Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of 45th Annual IEEE FOCS, pp. 32–41 (2004)

  21. Portugal, R., Santos, R.A.M., Fernandes, T.D., Gonçalves, D.N.: The staggered quantum walk model. Quantum Inf. Process. 15(1), 85–101 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  22. Portugal, R.: Establishing the equivalence between Szegedy’s and coined quantum walks using the staggered model. Quantum Inf. Process. 15(4), 1387–1409 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  23. Wong, T.: Equivalence of Szegedy’s and coined quantum walks. Quantum Inf. Process. 16(9), 215 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  24. Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  25. Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91, 207901 (2003)

    Article  ADS  Google Scholar 

  26. Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92, 187902 (2004)

    Article  ADS  Google Scholar 

  27. Christandl, M., Datta, N., Dorlas, T.C., Ekert, A., Kay, A., Landahl, A.J.: Perfect transfer of arbitrary states in quantum spin networks. Phys. Rev. A 71, 032312 (2005)

    Article  ADS  Google Scholar 

  28. Burgarth, D., Bose, S.: Conclusive and arbitrarily perfect quantum state transfer using parallel spin chain channels. Phys. Rev. A 71, 052315 (2005)

    Article  ADS  Google Scholar 

  29. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics. III. Quantum Mechanics. Addison-Wesley, Reading (1965)

    MATH  Google Scholar 

  30. Zimborás, Z., Faccin, M., Kádár, Z., Whitfield, J., Lanyon, B., Biamonte, J.: Quantum transport enhancement by time-reversal symmetry breaking. Sci. Rep. 3, 02361 (2013)

    Article  ADS  Google Scholar 

  31. Lu, D., Biamonte, J.D., Li, J., Li, H., Johnson, T.H., Bergholm, V., Faccin, M., Zimborás, Z., Laflamme, R., Baugh, J., Lloyd, S.: Chiral quantum walks. Phys. Rev. A 93, 042302 (2016)

    Article  ADS  Google Scholar 

  32. Stancil, D.D., Prabhakar, A.: Spin Waves: Theory and Applications. Springer, New York (2009)

    MATH  Google Scholar 

  33. Thompson, K.F., Gokler, C., Lloyd, S., Shor, P.W.: Time independent universal computing with spin chains: quantum plinko machine. New J. Phys. 18, 073044 (2016)

    Article  ADS  Google Scholar 

  34. Kontorovich, V.M., Tsukernik, V.M.: Spiral structure in one-dimensional chain of spins. Sov. Phys. JETP 25, 960–964 (1967)

    ADS  Google Scholar 

  35. Derzhko, O., Verkholyak, T., Krokhmalskii, T., Büttner, H.: Dynamic probes of quantum spin chains with the Dzyaloshinskii–Moriya interaction. Phys. Rev. B 73, 214407 (2006)

    Article  ADS  Google Scholar 

  36. Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover Publications, New York (1931)

    MATH  Google Scholar 

  37. Schumacher, B., Werner, R.F.: Reversible quantum cellular automata (2004). Preprint arxiv:quant-ph/0405174

  38. Jordan, P., Wigner, E.: Über das Paulische Äquivalenzverbot. Zeitschrift für Physik 47, 631–651 (1928)

    Article  ADS  Google Scholar 

  39. Lang, S.: Algebra. Springer, New York (2002)

    Book  Google Scholar 

  40. Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  41. DiVincenzo, D.P.: Two-bit gates are universal for quantum computation. Phys. Rev. A 51, 1015–1022 (1995)

    Article  ADS  Google Scholar 

  42. Postnikov, M.M.: Lie Groups and Lie Algebras. Mir Publishers, Moscow (1986)

    Google Scholar 

  43. Vlasov, A.Y.: Permanents, bosons and linear optics. Laser Phys. Lett. 14, 103001 (2017)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author is grateful to anonymous referees for corrections and suggestions to initial version of the article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexander Yu. Vlasov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vlasov, A.Y. Effective simulation of state distribution in qubit chains. Quantum Inf Process 17, 269 (2018). https://doi.org/10.1007/s11128-018-2036-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-018-2036-1

Keywords

Navigation